Subgradients and the Subdifferential of a Real-Valued Function on a Subset of $\mathbb{R}^n$

In the setting of Convex Functions on $\mathbb{R}^n$: Standing Notation, let $C\subseteq\mathbb{R}^{n}$, let $f:C\to\mathbb{R}$ be a function, and let $x\in C$.

(i) Subgradient. A point $a\in\mathbb{R}^{n}$ is a subgradient of $f$ at $x$ if

$$f(y)\ge f(x)+a\cdot(y-x)\qquad\text{for every }y\in C.$$

(ii) Subdifferential. The subdifferential of $f$ at $x$ is the set

$$\partial f(x)=\bigl\{a\in\mathbb{R}^{n}:\ a\text{ is a subgradient of }f\text{ at }x\bigr\}$$

of the subgradients of $f$ at $x$.